Science: What Was Grothendieck Thinking About?

The University of Montpellier has placed on line 18,000 pages of unedited notes made by the mathematical genius with his singular trajectory. The objective: to encourage new discoveries.

Along the manuscript pages are aligned mysterious equations, concentric schemes, all scribbled in a lively and compact handwriting.

photo: Maugendre-FishEye/Andia.fr

Clearly, one understands nothing. The “sheaves, stratified spaces” or the “conic neighborhoods” are meaningful to — practically — nobody. And no more meaningful, the “tensorial categories” or the “system of pseudo-lines”. But, we must admit, to plunge into the manuscript archive of the illustrious French researcher Alexandre Grothendieck, who died in November of 2014, after his tumultuous and battering life, procures, even for a philistine, a hefty dose of excitement and a dizzying impression of penetration into an uncommon mind. Since 16h30 on Wednesday 10 May, the University of Montpellier has put on line the nearly 18,000 unpublished notes made by this mathematical genius, designated as one of the greatest mathematicians of the 20th century. And with a rather insane hope: that of finding in this labyrinthine manuscript — Grothendieck spoke of them as his “scribblings” — the keys to new scientific advances.

All these documents come from a collection of 28,000 sheets drafted between the years 1950 and 1990 and bequeathed by the master to his pupil Jean Malgoire, still professor at the University of Montpellier. The half-dozen cartons have long slept in a storage room on the first floor of that establishment, until an agreement was finally reached with the children of the mathematician, who retain the physical property of the manuscripts but have ceded the publication rights to the university. “There are nuggets and pearls in this content, no doubt,” enthuses Jean-Michel Marin, director of the Department of Mathematics.

“Reconciling algebra, which proves, with geometry, which exhibits”

While this journey in the meanders of "grothendieckienne" thought is open to all, it will be understandable only to a few. On page after page, there are aligned mysterious equations, concentric patterns, sequences of reasoning, erasures, all scribbled in a lively and compact handwriting, sometimes on simple sheets of printer paper... A world inaccessible to ordinary mortals . Each handwritten page requires some ten hours of work for a skilled specialist in "algebraic geometry", the specialty that Alexander Grothendieck has revolutionized. “Exploiting his notes will take years for the scientific community,” confirms Jean-Michel Marin. “There are only a few hundred people in the world who can understand them. They contain not only unpublished results, but also tools to decipher his thinking.”

A tortuous and brilliant thought whose aim has always been, as explained by the journalist Philippe Douroux, who devoted a book [1] to Grothendieck’s life, “to reconcile algebra, which proves, with geometry, which exhibits.” This unachieved ambition was the guiding thread of a tormented destiny that saw Alexandre Grothendieck pass from the glory of a Fields medal (the Nobel Math) to an end of his existence walled up in the paranoid solitude of his house in Lasserre (Ariège ), lost in the foothills of the Pyrenees. Undoubtedly, these friends felt, he never succeeded in overcoming the trauma of a “terrible childhood.”

Alexandre was born in Berlin in 1928 of the love of a Ukrainian revolutionary of Jewish origin and a German anarchist. He was four years old when he was entrusted to a Hamburg family. His parents fled Nazi Germany and went to fight alongside the Spanish republicans. In 1939, he joined them in the south of France. But not for long. His father died in Auschwitz in 1942. Alexander was interned with his mother at the camp of Rieucros in Lozère. It was in those miserable barracks that the 11-year-old discovered, alone, the pleasure of Mathematics. By dint of intuition, he made the link between the circumference of a circle and its diameter. "You have to multiply by three," he told himself. To a few decimal places (3.14), that’s right. The young Grothendieck was then hidden at the Cévenol college of Chambon-sur-Lignon, where he obtained his baccalaureate, then a bachelor’s degree at the faculty of Montpellier. Spotted by a teacher, he followed the courses given by mathematicians Laurent Schwartz and Jean Dieudonné. The two scientists confided to him "fourteen problems" that they had not been able to solve. Grothendieck, hardly more than 20 years of age, succeeded in solving them all in a few months. His legend was in the making.

He joined the group of mathematicians “Nicolas Bourbaki” and won the Fields Medal in 1966. But Alexander Grothendieck did not go to pick it up. The man always despised honors and refused all prizes. His personality, without concession, was outside normal bounds. A prodigious power of work, paired with a mad dog side. “His life will be composed of ruptures,” sums up Philippe Douroux. Some conceptual ruptures, but also some social, familial, and political ruptures. A militant ecologist and antimilitarist, Grothendieck, the father of five children, remained stateless until 1971, refusing to do his military service. In 1970, he even slammed the door of the Institute of Advanced Studies in Science [2], after learning that it was receiving a grant from the Ministry of Defense ... Critic of "scientism", involved in the intellectual currents that advocated the return to the Land, Grothendieck gradually moved away from the scientific world, before spending the last 24 years of his life as a recluse in his home in the Ariège. But, during this inner journey, he never ceased to put his mathematical intuitions on paper. In addition to the 18,000 pages of notes published by the university, another 65,000 pages were written during the last two decades of his life: A last gold mine left as the inheritance of the unclassifiable Grothendieck.

[1] Alexandre Grothendieck. In the footsteps of the last genius of mathematics. Allary Editions, 266 pages, 18,90 euros.